calculus and analysis - Teaching Mathematica more about DiracDelta and KroneckerDelta

As the documentation and some experimentation indicates, Mathematica contains little information about representations of the DiracDelta and KroneckerDelta functions.

Some examples which are not recognized in Mathematica include the identities:

1. Limit[a/(x^2 + a^2), a -> 0] = DiracDelta[x]


2. Integrate[Exp[I k x], {x, -∞, ∞}, Assumptions -> k ∈ Reals]


3. Sum[Exp[I k n], {n, -∞, ∞}] = KroneckerDelta[k]


4. Integrate[h[n, x] h[m, x], {x, -∞, ∞}] = KroneckerDelta[n, m]


5. Sum[h[n, x] h[n, y], {n, 0, ∞}] = DiracDelta[x - y]

Where the set of functions h[n, x] is orthonormal.

I want to implement these kinds of relations in Mathematica. What would be the programmatic way to do this?